brianlok547

1. 農莊內飼養一批小雞，已知5%的小雞受到一種病毒感染。

(a)小雞輪流接受檢查，每次一隻，檢查後把牠放回小雞群，直至找到一隻受感染的小雞。求至少要檢查五次的概率。
(b)檢查農莊內５隻小雞，求只有３隻小雞受感染的概率。

2. 兩燈塔A和B獨立發射訊號。假設每小時的頻率分別為8次和6次，且它們發放的訊號數目各自依循泊松分佈。求15分鐘內兩燈塔總共射出三次訊號的概率。

3. 一學生需要回答六條問題。他答對任何一題的概率是0.7。若他答對n條問題，他會得到(15n+10)分。

(a)求該名學生所得分數的期望值。
(b)求第四題為首次答對的概率。
(c)求他的分數會低過期望值的概率。

brianlok547

麻煩替我詳細解答>.<

syms5104

唔好介意我用英文回你
Let X be the random variable representing the number of chickens infected
(a)1-0.05-0.95x0.05-095x0.95x0.05-0.95x0.95x0.95x0.05
=0.81450625
(b) X~B(5,0.05)
5C3 x 0.05x0.05x0.05 x 0.95x0.95
=0.001128125

syms5104

2 LET X and Y be the random variable representing the number of signals emitted by beacon A and B per 15 minutes respectively
X~Po(2)
Y~Po(1.5)
the required probabillity
=P(X=0)x P(Y=3)+ P(X=1) x P(Y=2) + P(X=2) x P( Y=1)+ P(X=3) x P(Y=0)
=0.215785469

syms5104

3 Let X be the random variable representing the number of questions that are correctly answered
(a) X~B(6,0.7)
The required probability
15(6x0.7)+10
=73
(b) X~G(0.7)
The required probability
0.3x0.3x0.3x0.7
=0.01789
(c)
The required probability
P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)
=0.579825

brianlok547

可否詳細列出P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)? (Q3)
和X~P0(2)和Y~P0(1.5)中的2和1.5怎樣出來? (Q2)

cs16744

2=8/4
1.5=6/4

cs16744

1(a)最少5次=頭4次要抽唔到
Let X be random variable representing infected chickens.
P(X>=5)
=1-P(X<=4)
=1-P(X=1)-P(X=2)-P(X=3)-P(X=4)
=1-0.05-0.05(0.95)-0.05(0.95)ˆ2-0.05(0.95ˆ3

(b)5隻雞入邊要有3隻=binomial distribution
P(X=3)
=5C3 (0.05)ˆ3 (0.95)ˆ2

2.每個鐘=60分鐘,人地要計每15分鐘,所以要計番每15分鐘既平均數
A=8/4=2次/15分鐘
B=6/4=1.5次/15分鐘

P(A=3&B=0)+P(A=2&B=1)+P(A=1&B=2)+P(A=0&B=3)
=e^(-2)*e^(-1.5)*[2^(3)/3! +1 +2^(2)*1.5/2! + 2*1.5^(2)/2! + 1.5^(3)/3!]